A357644 Number of integer compositions of n into parts that are alternately unequal and equal.

1, 1, 1, 3, 4, 7, 8, 13, 17, 25, 30, 44, 58, 77, 98, 142, 176, 245, 311, 426, 548, 758, 952, 1319, 1682, 2308, 2934, 4059, 5132, 7087, 9008, 12395, 15757, 21728, 27552, 38019, 48272, 66515, 84462, 116467, 147812, 203825, 258772, 356686, 452876, 624399, 792578

Offset: 0

Author

Gus Wiseman, Oct 14 2022

Keywords

• nonn

Examples

			The a(1) = 1 through a(7) = 13 compositions:
  (1)  (2)  (3)   (4)    (5)    (6)     (7)
            (12)  (13)   (14)   (15)    (16)
            (21)  (31)   (23)   (24)    (25)
                  (211)  (32)   (42)    (34)
                         (41)   (51)    (43)
                         (122)  (411)   (52)
                         (311)  (1221)  (61)
                                (2112)  (133)
                                        (322)
                                        (511)
                                        (2113)
                                        (3112)
                                        (12211)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Without equal relations we have A000213, equal only A027383.

Even-length opposite: A003242, ranked by A351010, partitions A035457.

The version for partitions is A351006.

The opposite version is A357643, partitions A351005.

A011782 counts compositions.

A357621 gives half-alternating sum of standard compositions, skew A357623.

A357645 counts compositions by half-alternating sum, skew A357646.

Cf. A001590, A029862, A035544, A097805, A122129, A122134, A122135, A351003, A351004, A351007, A357136, A357641.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],And@@Table[#[[i]]==#[[i+1]],{i,2,Length[#]-1,2}]&&And@@Table[#[[i]]!=#[[i+1]],{i,1,Length[#]-1,2}]&]],{n,0,10}]

Extensions

More terms from Alois P. Heinz, Oct 19 2022